I would expect that it is up to T32 Program Director (PI) to determine whether the mentor's research constitutes the "supervised research training" of the program.  If it does, then the student is refusing to be trained.
From the NIH policy statement:
"Trainees appointed to the training program must have the opportunity to carry out supervised biomedical or behavioral research with the primary objective of developing or extending their research skills and knowledge in preparation for a research career... Research trainees are expected to devote full time to the proposed research training, devoting at least 40 hours per week to the program."

-----Original Message-----
From: Research Administration List [mailto:xxxxxx@hrinet.org] On Behalf Of Bloomberg, Robert
Sent: Wednesday, July 01, 2009 2:53 PM
To: xxxxxx@hrinet.org
Subject: Re: [RESADM-L] T32 predoc working on mentor's research

He refused.

Robert P. Bloomberg
Director, Office of Grants & Contracts
Tufts Medical Center
800 Washington Street (Tupper 10)
Boston, MA  02111
Tel.   617.636.1142
Fax.  617.636.8568

> -----Original Message-----
> From: Research Administration List [mailto:xxxxxx@hrinet.org] On Behalf
> Of Charlie Hathaway
> Sent: Wednesday, July 01, 2009 3:51 PM
> To: xxxxxx@hrinet.org
> Subject: Re: [RESADM-L] T32 predoc working on mentor's research
>
> This kid should prepare for the LSAT.
>
> > I have a T32 question.  Usually predocs appointed to our T32 training
> > grants work on their mentor's research project and develop their
> > dissertation as an offshoot of the mentor's research, based what they
> > learn from the mentor. However, we have a predoc appointed to a T32 who
> > is refusing to work on his mentor's research. The student states that
> > the T32 appointment guarantees protected time only for dissertation
> > research and that working on the mentor's project isn't protected time.
> > I can't find any policies about protected time on T32s.  Thoughts?
> >
> >
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>
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 Instructions on how to use the RESADM-L Mailing List, including
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 Instructions on how to use the RESADM-L Mailing List, including
 subscription information and a web-searchable archive, are available
 via our web site at http://www.hrinet.org (click on "Listserv Lists")
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